for suitable \(c \in \mathbb {R}\) . Note that \(\frac{\gamma -2}{\gamma -1}\) -Hölder estimates are true even for subsolutions, and hold up to the boundary: this shows that superquadratic problems enjoy some properties that are rather unnatural for elliptic equations. Besides, solutions to uniformly elliptic (quasilinear) equations having subquadratic (subnatural) growth in the gradient are known to be Hölder continuous when \(q > \frac{N}{2}\)  (see e.g. [1]}, [2]}), and this classical fact leans on the perturbative nature of \(H(Du)\)  when \(\gamma < 2\) .
